COMPUTING POINTS ON BIELLIPTIC MODULAR CURVES OVER FIXED QUADRATIC FIELDS

Abstract We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves 
$X_0(N)$
 over fixed quadratic fields. We study 
$X_0(N)(\mathbb {Q}(\sqrt {d}))$
 for 
$N \in \{ 53,61,65,79,83,89,101,131 \}$
 and 
${\lvert d \rvert < 100}$
 .


Introduction
There has been a lot of recent interest in computing low-degree points on modular curves, and in particular in computing quadratic points on the curves X 0 (N ).Computing such points gives much insight into the arithmetic of elliptic curves and has direct applications in the resolution of Diophantine equations (see [8, p. 888] or [10] for such examples).
As we range over all quadratic fields, a curve X 0 (N ) will either have finitely many or infinitely many quadratic points.For those curves X 0 (N ) that have finitely many quadratic points, these points have been computed in many cases, such as when the genus of X 0 (N ) is ≤ 5, or when X 0 (N ) is bielliptic [5,14,12].If X 0 (N ) has genus ≥ 2 and has infinitely many quadratic points (so that X 0 (N ) is either hyperelliptic, or bielliptic with an elliptic quotient of positive rank over Q), a geometric description of all the quadratic points has been given in these cases [5,6,12].
There are precisely 10 values of N such that the modular curve X 0 (N ) is bielliptic with an elliptic quotient of positive rank [3, pp. 26-28].For two of these values of N , namely 37 and 43, the methods we present will not work (see Remark 2.2), and so we will consider the remaining eight values of N , which are N ∈ N := {53, 61, 65, 79, 83, 89, 101, 131}.
For each N ∈ N the elliptic curve X + 0 (N ) = X 0 (N )/w N has rank 1 over Q.In [5,12] it is proven that every quadratic point on X 0 (N ) arises as the pullback of a rational point on X + 0 (N ) (via the natural degree 2 quotient map).However, this classification does not describe X 0 (N )(K) for a given quadratic field K.The purpose of this paper is to introduce a Mordell-Weil sieve that can be used to check, for N ∈ N , whether X 0 (N )(K) = X 0 (N )(Q) for a given quadratic field K.The sieve uses information on the splitting behaviour of primes in K together with the structure of the Mordell-Weil group of X + 0 (N )(Q) modulo these primes.The sieve builds on ideas present in the author's work in [10, pp. 338-340].We prove the following result.Although we have considered integers d satisfying |d| < 100 here, there are no apparent obstructions to proving analogous results for any integer d.
For certain (but not all) integers d, the results of Theorem 1 could be achieved by applying [13,Theorem 1.1] or some of the techniques described in [2].In Section 3, we compare (for N = 53) our results with those one can obtain by applying [13,Theorem 1.1], and use this to provide an example of a curve that violates the Hasse principle.
We note that results of a similar nature to Theorem 1 (obtained using different techniques) are proven in [11] for hyperelliptic curves X 0 (N ).
The Magma [4] code used to support the computations in this paper is available at https://github.com/michaud-jacobs/bielliptic.

A Mordell-Weil sieve
In this section we present a Mordell-Weil sieve and apply it to prove Theorem 1.
We first describe how to obtain a suitable model for X 0 (N ) for N ∈ N .Let g denote the genus of the modular curve X 0 (N ).We start by computing a basis f 1 , . . ., f g of cusp forms for S 2 (Γ 0 (N )) with integer Fourier coefficients such that the Atkin-Lehner involution w N satisfies w N (f 1 ) = f 1 and w N (f i ) = −f i for 2 ≤ i ≤ g (we refer to such a basis as a diagonalised basis).For each N ∈ N the curve X 0 (N ) is non-hyperelliptic of genus > 2 and we may obtain a nonsingular model for X 0 (N ) over Q in P g−1 x 1 ,...,xg as the image of the canonical embedding on the cusp forms f 1 , . . ., f g .The details of this (standard) procedure are described in [9, pp. 17-38], and the Magma code we used to do this is adapted from [14].
The Atkin-Lehner involution w N on this model is then given by the map (x 1 : We denote by ψ : X 0 (N ) → X + 0 (N ) the degree 2 map induced by quotienting by w N .In each case, we found that the projection map onto the coordinates x 2 , . . ., x g had degree 2 and image X + 0 (N ) (and not some quotient of X + 0 (N )), so that the map ψ is given by We in fact then obtained a Weierstrass model for X + 0 (N ) and composed ψ with this transformation (see the example in Section 3).The reason for using a diagonalised model for X 0 (N ) is twofold.First, it forces the coordinates of a quadratic point to be of a certain shape, as we see below.Secondly, it greatly speeds up the computations we perform in the sieving step.
Let K = Q( √ d) be a quadratic field and write σ for the generator of Gal(K/Q).Suppose that P ∈ X 0 (N )(K)\X 0 (N )(Q) (equivalently, P is a non-cuspidal quadratic point).In projective coordinates, we may write , and , As discussed in the introduction, thanks to the work of Box and Najman-Vukorepa in [5,12], we know that ψ(P ) = ψ(P σ ) ∈ X + 0 (N )(Q), or equivalently, w N (P ) = P σ .It follows that with b 1 = 0, and we may assume that gcd(b 1 , a 2 , . . ., a g ) = 1 by rescaling if necessary.
We now present the sieve in the case that N = 65.In the case N = 65 we will need to adapt the sieve slightly, and we discuss this case in the proof of Theorem 1 below.For each N = 65, we have that X + 0 (N )(Q) ∼ = Z, and we let R denote a generator of the Mordell-Weil group, so that Let ℓ be a prime of good reduction for our models of X 0 (N ) and X + 0 (N ), and consider the following commutative diagram, where ∼ denotes reduction mod ℓ, or a prime of K above ℓ: X 0 (N ) By commutativity we have that ψ ℓ ( P ) = m• R, so that P ∈ ψ * ℓ (m• R).Write G ℓ for the order of R in the group X + 0 (N )(F ℓ ).Then . There are three cases: will not consist of a pair of distinct points defined over F ℓ , and so m ≡ m 1 (mod G ℓ ).
(ii) The set ψ * ℓ (m 1 • R) consists of a pair of points defined over F ℓ 2 (with each point not defined over F ℓ ).
If ℓ splits or ramifies in K, then ψ * ℓ (m • R) will not consist of a pair of points defined over F ℓ 2 , and so m ≡ m 1 (mod G ℓ ).
(iii) The set ψ * ℓ (m 1 • R) consists of a single point defined over F ℓ .
Verifying the splitting behaviour of the prime ℓ in cases (i) and (ii) leaves us with a list of possible values for m (mod G ℓ ).
We may then repeat this process with a list of primes ℓ 1 , . . ., ℓ s .For each 1 ≤ i ≤ s we obtain a list of possibilities for m (mod G ℓ i ).This gives a system of congruences that we may solve using the Chinese remainder theorem to obtain a list of possibilities for m (mod lcm(G ℓ i ) 1≤i≤s) ).If no solution exists to this system of congruences then we obtain a contradiction and conclude that X 0 (N )(K) = X 0 (N )(Q).

Proof of Theorem 1 for
We computed the preimages ψ * (t • R) for t ∈ Z with |t| ≤ 5 and verified the field of definition of the points we obtained.For each d ∈ D N we found a pair of quadratic points in For the converse, we suppose that |d| < 100 with d / ∈ D N and aim to prove that We applied the Mordell-Weil sieve described above with the following (ordered) choice of primes (we discuss this choice in Remark 2.1): In each case this led to a contradiction.
Proof of Theorem 1 for N = 65.The proof of the theorem in this case is very similar to the case N = 65.The key difference is that X + 0 (N )(Q) ∼ = Z ⊕ Z/2Z.We write Q for the 2-torsion point and choose a point R such that any point in X + 0 (N )(Q) may be expressed as m • R + n • Q for some m ∈ Z and n = 0 or 1.For our choice of R, we found that ψ * (−R) and ψ * (−2R) consisted of pairs of quadratic points defined over Q( √ −1) and Q( √ −79) respectively, proving one direction of the theorem.
For the converse, let d / ∈ D N be such that Q( √ d) is a quadratic field and |d| < 100.Suppose, for a contradiction, that there exists a point P ∈ In the first case, we apply the sieve exactly as in the proof for N = 65 (with the same choice of primes) to achieve a contradiction.In the second case, we again apply the sieve in the same way, except that we work with the point m•R+Q instead.To be precise, for each prime ℓ, we have that P ∈ ψ * ℓ (m • R + Q), and so we compute By considering each preimage and the splitting behaviour of ℓ in the quadratic field Q( √ d) we obtain a list of possibilities for m (mod G ℓ ).As in the previous case, we achieved a contradiction for each d.
The total computation time for the proof of Theorem 1 was 2500 seconds running on a 2200MHz AMD Opteron.
Remark 2.1.We discuss the choice of primes L used in the proof of the theorem.We start by choosing the primes that ramify as these usually eliminate the greatest number of possibilities for m (mod G ℓ ).We then choose primes ℓ such that the values G ℓ are small and share many prime factors.There are two reasons for doing this.First, when solving each system of congruences we are more likely to obtain fewer solutions, and ultimately a contradiction.Secondly, we avoid (or reduce the likelihood) of a combinatorial explosion, since the lowest common multiple of the G ℓ can grow very quickly if the primes ℓ are not chosen carefully.We note that the largest prime ℓ we in fact ended up reaching was ℓ = 593 in the case N = 101 and d = 31.Remark 2.2.As discussed in the introduction, we have not considered the curves X 0 (37) or X 0 (43).The curve X 0 (37) is bielliptic with an elliptic quotient of positive rank, but it is also hyperelliptic, and therefore has two sources of infinitely many quadratic points, meaning the sieve we have presented would not work.The reason the sieve does not work for the curve X 0 (43) is due to the fact that X 0 (43) has a non-cuspidal rational point that is fixed by the Atkin-Lehner involution w 43 .The sieve cannot rule out the possibility that a quadratic point is equal to this non-cuspidal rational point.
Although we have presented this sieve for certain specific bielliptic modular curves X 0 (N ), the sieve could be suitably adapted to compute quadratic points on a wider range of curves.Indeed, it should even be possible to apply a similar sieve to compute quadratic points on any curve X with a degree 2 quotient of genus ≥ 1, if there are finitely many quadratic points on X not arising as pullbacks of rational points on this quotient, and that these have all been computed.Although, as in the case X 0 (43) discussed above, there may be obstructions to the sieving process succeeding.